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A Survey of Visualization Methods for Special Relativity

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A Survey of Visualization Methods for Special Relativity A Survey of Visualization Methods for Special Relativity Daniel Weiskopf1 1 Visualization Research Center (VISUS), Universität Stuttgart [email protected] Abstract This paper provides a survey of approaches for special relativistic visualization. Visualization techniques are classified into three categories: Minkowski spacetime diagrams, depictions of spa- tial slices at a constant time, and virtual camera methods that simulate image generation in a relativistic scenario. The paper covers the historical outline from early hand-drawn visual- izations to state-of-the-art computer-based visualization methods. This paper also provides a concise presentation of the mathematics of special relativity, making use of the geometric nature of spacetime and relating it to geometric concepts such vectors and linear transformations. 1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling, J.2 Physical Sciences and Engineering Keywords and phrases Special Relativity, Minkowski, Spacetime, Virtual Camera Digital Object Identifier 10.4230/DFU.SciViz.2010.289 1 Introduction Einstein’s special theory of relativity has been attracting a lot of attention from the general public and physicists alike. In 2005, the 100th anniversary of the publication of special relativity [7] was the reason for numerous exhibitions, popular-science publications, or TV shows on modern physics in general and Einstein and his work in particular. In special relativistic physics, properties of space, time, and light are dramatically different from those of our familiar environment governed by classical physics. We do not experience relativistic effects and thus do not have an intuitive understanding for those effects because we, in our daily live, do not travel at velocities close to the speed of light. Special relativity is usually described in terms of mathematical models such as spacetime and Lorentz transformations. Since special relativity has a strong geometric component, visualization can play a crucial role in making those geometric aspects visible without relying on symbolic notation. This paper gives an overview of different approaches for visualizing various aspects of special relativity to a range of different audiences. One application is the support of visual communication for a general public, for example, by means of illustrations in popular-science publications, exhibitions, or TV shows. Another audience are students because visualization can be used to improve the learning experience in high- school or university courses. For example, depictions help to motivate, interactive computer experiments allow for exploration and active participation, and visual explanations can enrich a symbolic description of mathematical ideas. A third group of people are experts in physics and relativity. Although they do not need visualization to learn and understand the mathematics of special relativity, visualization may engage them in a different way of thinking. Edwin F. Taylor, a renowned teacher of special relativity [33], expressed his experience with special relativistic visualization as follows [32, © D. Weiskopf; licensed under Creative Commons License NC-ND Scientific Visualization: Advanced Concepts. Editor: Hans Hagen; pp. 289–302 Dagstuhl Publishing Schloss Dagstuhl – Leibniz Center for Informatics (Germany) 290 A Survey of Visualization Methods for Special Relativity Section 6]: “[...] I have come to think about relativity quite differently [...] My current view of the subject is much more visual, more fluid, more process oriented, covering a wider range of phenomena [...] In short, both my professional life and my view of physics have been transformed [by relativistic visualization].” These different types of audiences have a quite varying background knowledge. Therefore, different visualization approaches may be employed for different audiences and purposes. This paper distinguishes three classes of basic approaches (see Section 2): A direct visualization of spacetime by Minkowski diagrams, a visualization of a subset of spacetime for a fixed time, and the simulation of images as taken by a fast moving camera. Technical aspects and algorithms for these different approaches are discussed in Sections 5–7. All three approaches have in common that they may benefit from interactive computer implementations that facilitate trial-and-error explorations through the user. This paper has several goals. First, it provides a survey of state-of-the-art computer-based methods for special relativistic visualization. The methods are roughly structured along the aforementioned classification of approaches. Because several alternative algorithms are available for the camera metaphor, these algorithms are further classified in subcategories (see Section 7). In addition, this paper provides a concise presentation of special relativity in Section 3, making use of the geometric nature of spacetime and relating it to geometric concepts such vectors and linear transformations. This paper also provides a historical outline of the development of special relativistic visualization (Section 4). 2 Types of Visualization Approaches This paper adopts a geometric point of view on special relativity. Key elements are the concepts of spacetime and Lorentz transformations, which relate different coordinate systems in spacetime. Section 3 explains these concepts on a mathematical level. Spacetime is the combination of 3D space and a single temporal dimension, leading to a joint 4D description. Physical experiments, even if they are only virtual, are represented in form of 4D spacetime coordinates. For example, a point-like object that moves through space and time leaves a trace in spacetime—a so-called worldline. Similarly, light rays can be represented as lines through spacetime. Therefore, spacetime and traces therein are sufficient to describe the physical scenarios that are relevant for this paper. Lorentz transformations represent changes between coordinate systems—transformations between different frames of reference. Due to the Lorentz transformation, observers in different frames of reference typically provide different coordinate descriptions for the very same physical object. In other words, both spatial and temporal positions are dependent on the reference frame—space and time are not addressed by absolute coordinates, but they are relative. The structure of spacetime and the Lorentz transformation can be derived from two postulates: the principle of relativity (i.e., physical laws are valid and unchanged in any inertial reference frame) and the invariance of the speed of light (i.e., the speed of light in vacuo has a finite and constant value, regardless of the reference frame). This derivation can be found in textbooks, such as [22]. The three visualization approaches discussed in this paper can be related to spacetime in the following ways. All approaches have in common a reduction of dimensionality of 4D spacetime. (a) Minkowki diagrams. Minkowski diagrams are spacetime diagrams. They depict spacetime by graphically representing both temporal and spatial dimensions in a single image. The dimensionality of the spatial domain is reduced to either one or two (by taking a slice D. Weiskopf 291 through space), which leads to a total number of two or three dimensions for the spacetime diagram. In this way, graphical representations of the 2D or 3D diagram are feasible in an image. The advantage of Minkowski diagrams is their direct visualization of spacetime itself—Minkowski diagrams are the visual pendant to the mathematical geometry of special relativity. Figures 1 and 2 show typical examples of Minkowski diagrams. (b) Spatial slices. Another way of reducing dimensionality is to construct a spatial slice of constant time. This slicing corresponds physically to a simultaneous measurement of positions in 3D space. Time and simultaneity depend on the frame of reference, i.e., a spatial slice is always defined with respect to a reference frame. Spatial slices are a natural metaphor because they model measurements in 3D space for a “frozen” time. Figure 3 provides an example of several spatial slices taken at different times. (c) Virtual camera model. The virtual camera model simulates a physical experiment: what kind of image would a camera produce in a special relativistic setting? This approach simulates what we would see and, therefore, is the special relativistic analog of standard image synthesis by rendering non-relativistic scenes. Figure 4 illustrates the virtual camera view for high-speed travel toward the Brandenburg Gate. In a non-relativistic setting, where the speed of light is assumed to be infinite, approaches (b) and (c) are identical. Special relativity, however, requires us to make a clear distinction between seeing and measuring. Measurements are made at sample points simultaneously with respect to the reference frame of the observer. In contrast, seeing is based on the photons that arrive simultaneously at the camera of the observer. These photons are usually not emitted simultaneously (with respect to the observer’s reference frame) due to the finite speed of light. Following [41], approaches (a) and (b) can be regarded exocentric visualization, which present an outside view, whereas approach (c) can be considered an egocentric visualization, which is produced from the perspective of the user. 3 Elements of Special Relativity This section provides a brief introduction to the mathematics of special relativity, discussing the concepts of spacetime, Lorentz
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